Sums-of-Squares Formulas

An Application of Hermitian K-Theory: Sums-of-Squares Formulas

By using Hermitian K-theory, we improve D. Dugger and D. Isaksen’s condition (some powers of 2 dividing some binomial coefficients) for the existence of sums-of-squares formulas. 2010 Mathematics Subject Classification: 19G38; 11E25; 15A63

Etale Homotopy and Sums-of-squares Formulas

This paper uses a relative of BP -cohomology to prove a theorem in characteristic p algebra. Speci cally, we obtain some new necessary conditions for the existence of sums-of-squares formulas over elds of characteristic p > 2. These conditions were previously known in characteristic zero by results of Davis. Our proof uses a generalized etale cohomology theory called etale BP2.

Exact and Asymptotic Evaluation of the Number of Distinct Primitive Cuboids

We express the number of distinct primitive cuboids with given odd diagonal in terms of the twisted Euler function with alternating Dirichlet character of period four, and two counting formulas for binary sums of squares. Based on the asymptotic behaviour of the sums of these formulas, we derive an approximation formula for the cumulative number of primitive cuboids.

A Short Proof of Milne’s Formulas for Sums of Integer Squares

Infinite Families of Exact Sums of Squares Formulas, Jacobi Elliptic Functions, Continued Fractions, and Schur Functions

In this paper we derive many infinite families of explicit exact formulas involving either squares or triangular numbers, two of which generalize Jacobi’s 4 and 8 squares identities to 4n2 or 4n(n + 1) squares, respectively, without using cusp forms. In fact, we similarly generalize to infinite families all of Jacobi’s explicitly stated degree 2, 4, 6, 8 Lambert series expansions of classical t...